Figure REF  shows the absolute difference between the true bias and the various estimates, where for the direct estimate and the corrected estimate we show the results depending on \(n\) , the amount of data for which we have both \(y\)  and \(a\) . The boxplots are obtained from running the experiment 100 times.
We can see that the corrected estimate significantly improves over the naive estimate (e.g., the mean absolute difference, averaged over the 100 runs, is 0.0831 for the naive estimate and 0.0461 for the corrected estimate
with
\(n=250\) ).
Furthermore, the corrected estimate consistently outperforms the direct estimate. When only \(n=100\)  points of common data over \((y,a)\) 
are used,
the mean absolute difference for the corrected estimate is 0.054 while for the direct estimate it is 0.0837.
In this experiment, the mean true bias
is 0.14, the mean error of the label and the attribute classifier is 0.18 and 0.25, respectively, and the mean violation of the independence assumption according to the measure proposed in [1]} is \(0.017\) .
